First, I'll recommend that you check out Transatlantic Encounters: American Indians in Britain, 1500-1776 by Alden T. Vaughan if you're interested in this topic. I'll be using it as my main source here. Now, onto your questions:

1. For the vast majority of Americans heading east across the Atlantic, it was not a matter of choice, especially early in the colonial era. Columbus brought back ten Taino to survive both as proof that he had successfully discovered inhabited land beyond the Atlantic and to be trained as interpreters for his next voyage. Four were left behind in Seville, presumably too ill to travel further. Six reached Madrid with Columbus, where they were baptized and given new names. Don Juan de Castilla, as one came to be called, became a servant of the royal court for two years before he died. Another, Don Diego Colon, was the only one of the ten to return with Columbus on his next voyage as the first fully trained translator. Over the course of his second voyage and the conquest of Hispaniola, Columbus shipped 600+ Taino slaves back to Spain. There were thousands of Native American slaves in Spain by the time the American slave trade was outlawed in the mid-1500s. In Portugal, there were fewer but they were still "a common sight," as Vaughan describes the situation, and in France there were a few hundred Native Americans by the 1600. Of these, very few ever saw their homeland again, even when freed by imperial decree.

2. Tisquantum was one of these captives. We was originally captured by Ferdinando Gorges as he returned from Virginia in 1605, to serve as an interpreter for a future English expedition to Norumbega - what's now New England along with two others. Tisquantum returned to Norumbega in the company of John Smith in 1614. Smith departed soon after, leaving Thomas Hunt in charge of establishing a trading post with the local people. But Hunt had other ideas. He abducted Tisquantum and others again and sold them as slaves in Malaga, Spain. He was quickly set free in Spain and allowed to travel to London, where he worked for a merchant named John Slany. He was eventually hired on as a guide for another expedition to North America, which he took in hopes of returning home. The expedition arrived in Newfoundland in 1618, and before he could sail south to return to Massachusetts, the ship made another trip back to England for supplies. Once they were back in Newfoundland, he was finally able to head home.

3. The actual leaders stayed home and sent delegates and diplomats. For example, instead of going to England himself, Powhatan sent Namontack in 1608, Matchumps in ~1610, and Tomocomo in 1616 (this one being part of the same voyage that brought Pocahontas to England). Assuming you weren't on a diplomatic mission, or being forced into perpetual servitude, you'd need some sort of European sponsor. There were missionary organization brought Native children and young adults to England to receive European education, though there were probably less than a 100 of such cases in the 1600s, and their ultimate fates are unclear. Some people also managed the voyage to Europe by getting hired onto the crews of ships heading back and forth - this would be particularly common among the Native whalers from southern New England into the 19th Century. A notable exception to the "the leaders don't go" are the cases when the leaders themselves were abducted; for example, when Denonville was governor of New France in the 1680s, he captured 50 delegates the Haudenosaunee (Iroquois Confederacy) had sent to a peace summit he hosted, which include a sizable portion of the Grand Council (there are 49 permanent seats and a variable number of temporary seats on the Council, so the number 50 here is probably not coincidental). The captive councilors became galley slaves in the War of the Grand Alliance, and only 13 returned to North America.

4. How you'd fit in and how you'd be treated depends wildly on the manner in which you made the voyage across the Atlantic. Most were slaves, as I said, and not well treated at all. In the mid-1570s, first Americans in England were an Inuit "family" (the man, Kalicho, was probably actually unrelated to the woman and her son, called Arnaq and Nutaaq, though those likely weren't their names being suspiciously close to the Inuit words for "Woman" and "Child") that were essentially treated as little better than zoo animals, carted around England to be gawked at and locked in rooms while observant Englishman patiently waited in hopes that they'd mate. Namontack was quite pleased with the treatment he received as a diplomat and may have been given an audience with King James (the historical record is a bit ambiguous on this matter). Donnacona, the leader of Stadacona (now Quebec City), were among the dozen or so Stadconans abducted during Cartier's second voyage in 1535; though a captive, he was fairly well treated and became a favored guest in the court of King Francis I.

5. As mentioned, most who made the voyage across the Atlantic did not have the opportunity to return. Donnacona and the others with them were supposed to return to Canada with Cartier, but the expedition was delayed for years and all but one - a young girl - died in the intervening time. According to Cartier, she chose to remain in France because she had married a Frenchman. I can't say much about how many Europeans today can trace their ancestry back to Native Americans brought to Europe during this time period, but based on genetic evidence in Iceland at least, a sizable percentage of the population appear to be descended to Native American ancestors 300+ years ago.

6. See #2.

7. A major recurring theme, famously espoused by the Tupi delegation to France in the 1500s and Joseph Brant (Mohawk) during the Revolutionary era, was disgust over the rampant inequality in European societies. The Tupi diplomat interviewed by Montaigne was shocked to see French in poverty while their king had such wealth. Brant decried the debtor's prisons, saying that Europeans should cease to call themselves Christians while they imprison the poor and that he'd rather die by the most severe torture his ancestors could devise than waste away for years in an English prison. Since you asked the long-lasting buildings of Europe, I'll quote Brant's assessment of the palaces of Europe: "a deformed piece of earth, swelled with pride, and assuming airs, that become none but the Spirit above." Of course, both of these Native societies have fairly low levels of economic inequality; someone from Mesoamerica, with their own brand of inequalities, would be unlikely to react the same way but I don't have any info about that (maybe one of the Mesoamericanist around here can help out with that). It wasn't all negative though. Brant, for example, was particular fond of the education he had received from European teachers. Others, like Samsom Occom, a Mohegan preacher who visited London in 1766, disliked the squalor and crowd of the city but praised the grandeur of the royal court - saying that if an earthly king could appear so glorious, how much more so must God be.

8. They certain knew what was up with when travelling. Tomocomo, for example, is noted to have kept careful count of how many days it took to reach England. The English found it odd that his count was double the actual number of days, but this was because the ship traveled night and day and traditionally the Powhatan did not travel at night, so the doubling was a consequence of noting that they were traveling twice as much as would be normal for the Powhatan.

9. If you were a slave, you stayed with whoever purchased you. If you were brought over as a curiosity or a diplomat, you stayed with some aristocrat who could afford to keep you around as a human pet or foreign dignitary, as the case may be. Donnacona lived with Cartier when he wasn't visiting with the king, for example.

10. Slaves weren't given any leeway; the social faux pas of captive curiosities only made them more amusing to their captors. Diplomats like Namontack were coached in courtly manners. There were a few people who went to Europe for their own reasons. Around 1675, a Nipmuck man named John Wampas somehow managed to get to England in order to represent himself in a land deal before the highest court in the realm. He came to London without friends, social standing or money and soon found himself in a debtor's prison for a year or two, until eventually King Charles agreed to hear his case and set him free.